Enhanced $B \to \mu \bar\nu$ Decay at Tree Level as Probe of Extra Yukawa Couplings

With no New Physics seen at the LHC, a second Higgs doublet remains attractive and plausible. The ratio ${\cal R}_B^{\mu/\tau} = {\cal B}(B \to \mu \bar\nu)/{\cal B}(B \to \tau \bar\nu)$ is predicted at 0.0045 in both the Standard Model and the type II two Higgs doublet model, but it can differ if extra Yukawa couplings exist in Nature, which we deem an experimental issue. Considering recent Belle update on $B \to \mu\bar\nu$, we show that in the general two Higgs doublet model, the ratio could be up by a factor of two, which can be probed by the Belle~II experiment with just a few ab$^{-1}$.